// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2008-2010 Gael Guennebaud <gael.guennebaud@inria.fr>
// Copyright (C) 2009 Benoit Jacob <jacob.benoit.1@gmail.com>
// Copyright (C) 2010 Vincent Lejeune
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.

#ifndef EIGEN_QR_H
#define EIGEN_QR_H

namespace Eigen {

namespace internal {
template<typename _MatrixType>
struct traits<HouseholderQR<_MatrixType>> : traits<_MatrixType>
{
	typedef MatrixXpr XprKind;
	typedef SolverStorage StorageKind;
	typedef int StorageIndex;
	enum
	{
		Flags = 0
	};
};

} // end namespace internal

/** \ingroup QR_Module
 *
 *
 * \class HouseholderQR
 *
 * \brief Householder QR decomposition of a matrix
 *
 * \tparam _MatrixType the type of the matrix of which we are computing the QR decomposition
 *
 * This class performs a QR decomposition of a matrix \b A into matrices \b Q and \b R
 * such that
 * \f[
 *  \mathbf{A} = \mathbf{Q} \, \mathbf{R}
 * \f]
 * by using Householder transformations. Here, \b Q a unitary matrix and \b R an upper triangular matrix.
 * The result is stored in a compact way compatible with LAPACK.
 *
 * Note that no pivoting is performed. This is \b not a rank-revealing decomposition.
 * If you want that feature, use FullPivHouseholderQR or ColPivHouseholderQR instead.
 *
 * This Householder QR decomposition is faster, but less numerically stable and less feature-full than
 * FullPivHouseholderQR or ColPivHouseholderQR.
 *
 * This class supports the \link InplaceDecomposition inplace decomposition \endlink mechanism.
 *
 * \sa MatrixBase::householderQr()
 */
template<typename _MatrixType>
class HouseholderQR : public SolverBase<HouseholderQR<_MatrixType>>
{
  public:
	typedef _MatrixType MatrixType;
	typedef SolverBase<HouseholderQR> Base;
	friend class SolverBase<HouseholderQR>;

	EIGEN_GENERIC_PUBLIC_INTERFACE(HouseholderQR)
	enum
	{
		MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
		MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
	};
	typedef Matrix<Scalar,
				   RowsAtCompileTime,
				   RowsAtCompileTime,
				   (MatrixType::Flags & RowMajorBit) ? RowMajor : ColMajor,
				   MaxRowsAtCompileTime,
				   MaxRowsAtCompileTime>
		MatrixQType;
	typedef typename internal::plain_diag_type<MatrixType>::type HCoeffsType;
	typedef typename internal::plain_row_type<MatrixType>::type RowVectorType;
	typedef HouseholderSequence<MatrixType,
								typename internal::remove_all<typename HCoeffsType::ConjugateReturnType>::type>
		HouseholderSequenceType;

	/**
	 * \brief Default Constructor.
	 *
	 * The default constructor is useful in cases in which the user intends to
	 * perform decompositions via HouseholderQR::compute(const MatrixType&).
	 */
	HouseholderQR()
		: m_qr()
		, m_hCoeffs()
		, m_temp()
		, m_isInitialized(false)
	{
	}

	/** \brief Default Constructor with memory preallocation
	 *
	 * Like the default constructor but with preallocation of the internal data
	 * according to the specified problem \a size.
	 * \sa HouseholderQR()
	 */
	HouseholderQR(Index rows, Index cols)
		: m_qr(rows, cols)
		, m_hCoeffs((std::min)(rows, cols))
		, m_temp(cols)
		, m_isInitialized(false)
	{
	}

	/** \brief Constructs a QR factorization from a given matrix
	 *
	 * This constructor computes the QR factorization of the matrix \a matrix by calling
	 * the method compute(). It is a short cut for:
	 *
	 * \code
	 * HouseholderQR<MatrixType> qr(matrix.rows(), matrix.cols());
	 * qr.compute(matrix);
	 * \endcode
	 *
	 * \sa compute()
	 */
	template<typename InputType>
	explicit HouseholderQR(const EigenBase<InputType>& matrix)
		: m_qr(matrix.rows(), matrix.cols())
		, m_hCoeffs((std::min)(matrix.rows(), matrix.cols()))
		, m_temp(matrix.cols())
		, m_isInitialized(false)
	{
		compute(matrix.derived());
	}

	/** \brief Constructs a QR factorization from a given matrix
	 *
	 * This overloaded constructor is provided for \link InplaceDecomposition inplace decomposition \endlink when
	 * \c MatrixType is a Eigen::Ref.
	 *
	 * \sa HouseholderQR(const EigenBase&)
	 */
	template<typename InputType>
	explicit HouseholderQR(EigenBase<InputType>& matrix)
		: m_qr(matrix.derived())
		, m_hCoeffs((std::min)(matrix.rows(), matrix.cols()))
		, m_temp(matrix.cols())
		, m_isInitialized(false)
	{
		computeInPlace();
	}

#ifdef EIGEN_PARSED_BY_DOXYGEN
	/** This method finds a solution x to the equation Ax=b, where A is the matrix of which
	 * *this is the QR decomposition, if any exists.
	 *
	 * \param b the right-hand-side of the equation to solve.
	 *
	 * \returns a solution.
	 *
	 * \note_about_checking_solutions
	 *
	 * \note_about_arbitrary_choice_of_solution
	 *
	 * Example: \include HouseholderQR_solve.cpp
	 * Output: \verbinclude HouseholderQR_solve.out
	 */
	template<typename Rhs>
	inline const Solve<HouseholderQR, Rhs> solve(const MatrixBase<Rhs>& b) const;
#endif

	/** This method returns an expression of the unitary matrix Q as a sequence of Householder transformations.
	 *
	 * The returned expression can directly be used to perform matrix products. It can also be assigned to a dense
	 * Matrix object. Here is an example showing how to recover the full or thin matrix Q, as well as how to perform
	 * matrix products using operator*:
	 *
	 * Example: \include HouseholderQR_householderQ.cpp
	 * Output: \verbinclude HouseholderQR_householderQ.out
	 */
	HouseholderSequenceType householderQ() const
	{
		eigen_assert(m_isInitialized && "HouseholderQR is not initialized.");
		return HouseholderSequenceType(m_qr, m_hCoeffs.conjugate());
	}

	/** \returns a reference to the matrix where the Householder QR decomposition is stored
	 * in a LAPACK-compatible way.
	 */
	const MatrixType& matrixQR() const
	{
		eigen_assert(m_isInitialized && "HouseholderQR is not initialized.");
		return m_qr;
	}

	template<typename InputType>
	HouseholderQR& compute(const EigenBase<InputType>& matrix)
	{
		m_qr = matrix.derived();
		computeInPlace();
		return *this;
	}

	/** \returns the absolute value of the determinant of the matrix of which
	 * *this is the QR decomposition. It has only linear complexity
	 * (that is, O(n) where n is the dimension of the square matrix)
	 * as the QR decomposition has already been computed.
	 *
	 * \note This is only for square matrices.
	 *
	 * \warning a determinant can be very big or small, so for matrices
	 * of large enough dimension, there is a risk of overflow/underflow.
	 * One way to work around that is to use logAbsDeterminant() instead.
	 *
	 * \sa logAbsDeterminant(), MatrixBase::determinant()
	 */
	typename MatrixType::RealScalar absDeterminant() const;

	/** \returns the natural log of the absolute value of the determinant of the matrix of which
	 * *this is the QR decomposition. It has only linear complexity
	 * (that is, O(n) where n is the dimension of the square matrix)
	 * as the QR decomposition has already been computed.
	 *
	 * \note This is only for square matrices.
	 *
	 * \note This method is useful to work around the risk of overflow/underflow that's inherent
	 * to determinant computation.
	 *
	 * \sa absDeterminant(), MatrixBase::determinant()
	 */
	typename MatrixType::RealScalar logAbsDeterminant() const;

	inline Index rows() const { return m_qr.rows(); }
	inline Index cols() const { return m_qr.cols(); }

	/** \returns a const reference to the vector of Householder coefficients used to represent the factor \c Q.
	 *
	 * For advanced uses only.
	 */
	const HCoeffsType& hCoeffs() const { return m_hCoeffs; }

#ifndef EIGEN_PARSED_BY_DOXYGEN
	template<typename RhsType, typename DstType>
	void _solve_impl(const RhsType& rhs, DstType& dst) const;

	template<bool Conjugate, typename RhsType, typename DstType>
	void _solve_impl_transposed(const RhsType& rhs, DstType& dst) const;
#endif

  protected:
	static void check_template_parameters() { EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar); }

	void computeInPlace();

	MatrixType m_qr;
	HCoeffsType m_hCoeffs;
	RowVectorType m_temp;
	bool m_isInitialized;
};

template<typename MatrixType>
typename MatrixType::RealScalar
HouseholderQR<MatrixType>::absDeterminant() const
{
	using std::abs;
	eigen_assert(m_isInitialized && "HouseholderQR is not initialized.");
	eigen_assert(m_qr.rows() == m_qr.cols() && "You can't take the determinant of a non-square matrix!");
	return abs(m_qr.diagonal().prod());
}

template<typename MatrixType>
typename MatrixType::RealScalar
HouseholderQR<MatrixType>::logAbsDeterminant() const
{
	eigen_assert(m_isInitialized && "HouseholderQR is not initialized.");
	eigen_assert(m_qr.rows() == m_qr.cols() && "You can't take the determinant of a non-square matrix!");
	return m_qr.diagonal().cwiseAbs().array().log().sum();
}

namespace internal {

/** \internal */
template<typename MatrixQR, typename HCoeffs>
void
householder_qr_inplace_unblocked(MatrixQR& mat, HCoeffs& hCoeffs, typename MatrixQR::Scalar* tempData = 0)
{
	typedef typename MatrixQR::Scalar Scalar;
	typedef typename MatrixQR::RealScalar RealScalar;
	Index rows = mat.rows();
	Index cols = mat.cols();
	Index size = (std::min)(rows, cols);

	eigen_assert(hCoeffs.size() == size);

	typedef Matrix<Scalar, MatrixQR::ColsAtCompileTime, 1> TempType;
	TempType tempVector;
	if (tempData == 0) {
		tempVector.resize(cols);
		tempData = tempVector.data();
	}

	for (Index k = 0; k < size; ++k) {
		Index remainingRows = rows - k;
		Index remainingCols = cols - k - 1;

		RealScalar beta;
		mat.col(k).tail(remainingRows).makeHouseholderInPlace(hCoeffs.coeffRef(k), beta);
		mat.coeffRef(k, k) = beta;

		// apply H to remaining part of m_qr from the left
		mat.bottomRightCorner(remainingRows, remainingCols)
			.applyHouseholderOnTheLeft(mat.col(k).tail(remainingRows - 1), hCoeffs.coeffRef(k), tempData + k + 1);
	}
}

/** \internal */
template<typename MatrixQR,
		 typename HCoeffs,
		 typename MatrixQRScalar = typename MatrixQR::Scalar,
		 bool InnerStrideIsOne = (MatrixQR::InnerStrideAtCompileTime == 1 && HCoeffs::InnerStrideAtCompileTime == 1)>
struct householder_qr_inplace_blocked
{
	// This is specialized for LAPACK-supported Scalar types in HouseholderQR_LAPACKE.h
	static void run(MatrixQR& mat, HCoeffs& hCoeffs, Index maxBlockSize = 32, typename MatrixQR::Scalar* tempData = 0)
	{
		typedef typename MatrixQR::Scalar Scalar;
		typedef Block<MatrixQR, Dynamic, Dynamic> BlockType;

		Index rows = mat.rows();
		Index cols = mat.cols();
		Index size = (std::min)(rows, cols);

		typedef Matrix<Scalar, Dynamic, 1, ColMajor, MatrixQR::MaxColsAtCompileTime, 1> TempType;
		TempType tempVector;
		if (tempData == 0) {
			tempVector.resize(cols);
			tempData = tempVector.data();
		}

		Index blockSize = (std::min)(maxBlockSize, size);

		Index k = 0;
		for (k = 0; k < size; k += blockSize) {
			Index bs = (std::min)(size - k, blockSize); // actual size of the block
			Index tcols = cols - k - bs;				// trailing columns
			Index brows = rows - k;						// rows of the block

			// partition the matrix:
			//        A00 | A01 | A02
			// mat  = A10 | A11 | A12
			//        A20 | A21 | A22
			// and performs the qr dec of [A11^T A12^T]^T
			// and update [A21^T A22^T]^T using level 3 operations.
			// Finally, the algorithm continue on A22

			BlockType A11_21 = mat.block(k, k, brows, bs);
			Block<HCoeffs, Dynamic, 1> hCoeffsSegment = hCoeffs.segment(k, bs);

			householder_qr_inplace_unblocked(A11_21, hCoeffsSegment, tempData);

			if (tcols) {
				BlockType A21_22 = mat.block(k, k + bs, brows, tcols);
				apply_block_householder_on_the_left(A21_22, A11_21, hCoeffsSegment, false); // false == backward
			}
		}
	}
};

} // end namespace internal

#ifndef EIGEN_PARSED_BY_DOXYGEN
template<typename _MatrixType>
template<typename RhsType, typename DstType>
void
HouseholderQR<_MatrixType>::_solve_impl(const RhsType& rhs, DstType& dst) const
{
	const Index rank = (std::min)(rows(), cols());

	typename RhsType::PlainObject c(rhs);

	c.applyOnTheLeft(householderQ().setLength(rank).adjoint());

	m_qr.topLeftCorner(rank, rank).template triangularView<Upper>().solveInPlace(c.topRows(rank));

	dst.topRows(rank) = c.topRows(rank);
	dst.bottomRows(cols() - rank).setZero();
}

template<typename _MatrixType>
template<bool Conjugate, typename RhsType, typename DstType>
void
HouseholderQR<_MatrixType>::_solve_impl_transposed(const RhsType& rhs, DstType& dst) const
{
	const Index rank = (std::min)(rows(), cols());

	typename RhsType::PlainObject c(rhs);

	m_qr.topLeftCorner(rank, rank)
		.template triangularView<Upper>()
		.transpose()
		.template conjugateIf<Conjugate>()
		.solveInPlace(c.topRows(rank));

	dst.topRows(rank) = c.topRows(rank);
	dst.bottomRows(rows() - rank).setZero();

	dst.applyOnTheLeft(householderQ().setLength(rank).template conjugateIf<!Conjugate>());
}
#endif

/** Performs the QR factorization of the given matrix \a matrix. The result of
 * the factorization is stored into \c *this, and a reference to \c *this
 * is returned.
 *
 * \sa class HouseholderQR, HouseholderQR(const MatrixType&)
 */
template<typename MatrixType>
void
HouseholderQR<MatrixType>::computeInPlace()
{
	check_template_parameters();

	Index rows = m_qr.rows();
	Index cols = m_qr.cols();
	Index size = (std::min)(rows, cols);

	m_hCoeffs.resize(size);

	m_temp.resize(cols);

	internal::householder_qr_inplace_blocked<MatrixType, HCoeffsType>::run(m_qr, m_hCoeffs, 48, m_temp.data());

	m_isInitialized = true;
}

/** \return the Householder QR decomposition of \c *this.
 *
 * \sa class HouseholderQR
 */
template<typename Derived>
const HouseholderQR<typename MatrixBase<Derived>::PlainObject>
MatrixBase<Derived>::householderQr() const
{
	return HouseholderQR<PlainObject>(eval());
}

} // end namespace Eigen

#endif // EIGEN_QR_H
